The chain rule allows us to compute the derivative of a composite function. In other words suppose that:

y ~=~ f delim{[}{g(x)}{]}

for some functions f and g. For example:

y~=~sin(x^3)

In this case we can identify:

g ( x )~=~x^3

How do you compute the derivative in this case? The chain rule tells us that we compute the derivative in the following way:

{dy/dx}~=~{dg/dx}{df/dg}

Let’s compute some examples. Consider:

y~=~sin(x^2)

Here we have:

g(x) ~=~x^2~doubleright~{dg/dx}~=~2x

And

{df/dg}~=~{d/dg}sin(g)~=~cos(g)~=~cos(x^2)

Hence:

{dy/dx} ~=~ {dg/dx}{df/dg}~=~2xcos(x^2)

Here’s another example. Suppose you’re given:

y ~=~(1 ~+~2x)^100

In this case we have:

g ~=~ 1~+~2x~doubleright~{dg/dx}~=~2

Meanwhile:

{df/dg} ~=~100g^99~=~100(1~+~2x)^99

Putting everything together we get:

{dy/dx}~=~{dg/dx}{df/dg}~=~2(100)(1~+~2x)^99~=~200(1~+~2x)^99

Let’s do one more. It’s basically the same as the previous example with a minor twist:

y = {1/(3x~-~5)^7}

Proceeding as before we get:

g~=~3x-5~doubleright~{dg/dx}~=~3

Meanwhile

{df/dg} ~=~{d/dg}g^-7~=~-7g^-8

And so the final result is:

{dy/dx}~=~{dg/dx}{df/dg}~=~{-21/(3x~-~5)^8}